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Pre-AP Algebra 2
Unit 9 – Lesson 1 – Rational Exponents
Objectives:
Students will understand that a radical can be represented as a rational exponent
Students will be able to convert between radicals and rational exponents
Materials: Do Now and answers overhead; note-taking templates; practice worksheet; homework #9-
Time Activity
15 min DO NOW
- Student investigate rational exponents using their calculators
30 min Direct Instruction
Mathematicians wanted a way to write radical expressions – those with the root symbol – using
exponents, so that they could be worked with just like all other numbers. Here is a proof that shows why
it works.
Step Reason
√ √
By the definition of a square root
Define the radical as an unknown exponent
When multiplying, add exponents
Anything to the 1
st
power is itself
If the powers are equal, the exponents must be too
Solve for k. Thus, √
This proof could be repeated with any other n-th roots. You try! Prove that a
3
a
1
3
.
Examples
√
√
√
- √
What about more complex functions?
What about combinations?
( )
( )
20 min Pair Work
Practice worksheet
Homework #9-1: Rational Exponents
Pre-AP Algebra 2 Name: __________________________
DO NOW
DO NOW
Find the exact values for each of the following
What pattern do you see?
How could you find √
? Try it!
Pre-AP Algebra 2 Name: __________________________
Homework #9-
Homework #9-1: Rational Exponents
Part 1
- Find the exact, simplified value of each expression without a calculator. If you are stuck , try
converting between radical and rational exponential notation first, and then simplify.
Sometimes, simplifying the exponent (or changing a decimal to a fraction) is very helpful.
a.
3
1
125 b.
1 / 2
64 c.
1 / 6
d.
1 / 2
81 e.
1 / 5
32 f.
1 / 4
g.
3 / 2
4 h.(64)
2/
i. 8
5 /
j. 9
3/
k.
3/
l. 16
1.
m. 27
3
2
n. 125
3 2
o. 4
3
6
p. 5
2
q. 2
4
4
r. 3
5
5
- Simplify each expression completely.
a.
5/
1/
b. 5
2/ 3
1/ 2
c.
1/ 2
d.
2
2
1/ 2
e.
1/ 9
1/ 4
f.
3/ 4
3/ 4
4
Part 2: STAAR Practice
- A parabola has the function f(x) = 2(x + 3)
2
- It is translated to a new location, given by the
function g(x) = 2(x – 3)
2
- Describe the translation.
a. 6 left and 3 up b. 6 left and 3 down
c. 6 right and 3 down d. 6 right and 3 up
- What is the highest point on the function y = -(x – 5)
2
a.
b.
c.
d.
- A certain radioactive element decays over time according to the equation y A
t / 300
where A
is the number of grams present initially and t is the time in years. If 1000 grams were present
initially, how many grams will remain after 900 years?
a. 500 grams b. 250 grams c. 125 grams d. 62.5 grams
- Given the equation √
, which of the following represents in terms of?
a. b. c. d.
- Simplify:
x 7
x
2
2 x 35
a.
5 x 6
x
2
2 x 35
b.
5( x 5)
x
2
2 x 35
c.
5( x 3)
x
2
2 x 35
d.
5( x 7)
x
2
2 x 35
- Which figure best describes the graph of
a. circle b. ellipse c. parabola d. hyperbola
- The graph of the function was obtained from the graph of the
function using a transformation as shown above. Based on the
graph, which equation can be used to describe ( ) in terms of
a. ( ) ( ) b. ( ) ( )
c. ( ) ( ) d. ( ) ( )
Lesson Name: Rational Exponents Date: _______________ Student: ________________________
Concepts Examples
Mathematicians wanted a way to write radical expressions – those with the root
symbol – using exponents, so that they could be worked with just like all other
numbers. Here is a proof that shows why it works.
Step Reason
√ √
By the definition of a square root
Define the radical as an unknown
exponent
You try! Prove that a
3
a
1
3
.
Step Reason
By the definition of a cube root
Define the radical as an unknown
exponent
Examples
√
√
What about more complex functions?
Lesson Name: Rational Exponents Date: _______________ Student: ________________________
More Examples
What about combinations?
